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bsc-leon-vatthauer/public/Categories.Diagram.Coend.html

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<html><head><meta charset="utf-8"><title>Categories.Diagram.Coend</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="74" class="Keyword">using</a> <a id="80" class="Symbol">(</a><a id="81" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="89" class="Symbol">)</a>
<a id="91" class="Keyword">open</a> <a id="96" class="Keyword">import</a> <a id="103" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a> <a id="132" class="Keyword">using</a> <a id="138" class="Symbol">(</a><a id="139" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a><a id="148" class="Symbol">)</a>
<a id="151" class="Keyword">module</a> <a id="158" href="Categories.Diagram.Coend.html" class="Module">Categories.Diagram.Coend</a> <a id="183" class="Symbol">{</a><a id="184" href="Categories.Diagram.Coend.html#184" class="Bound">o</a> <a id="186" href="Categories.Diagram.Coend.html#186" class="Bound">ℓ</a> <a id="188" href="Categories.Diagram.Coend.html#188" class="Bound">e</a> <a id="190" href="Categories.Diagram.Coend.html#190" class="Bound">o′</a> <a id="193" href="Categories.Diagram.Coend.html#193" class="Bound">ℓ′</a> <a id="196" href="Categories.Diagram.Coend.html#196" class="Bound">e′</a><a id="198" class="Symbol">}</a> <a id="200" class="Symbol">{</a><a id="201" href="Categories.Diagram.Coend.html#201" class="Bound">C</a> <a id="203" class="Symbol">:</a> <a id="205" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="214" href="Categories.Diagram.Coend.html#184" class="Bound">o</a> <a id="216" href="Categories.Diagram.Coend.html#186" class="Bound">ℓ</a> <a id="218" href="Categories.Diagram.Coend.html#188" class="Bound">e</a><a id="219" class="Symbol">}</a> <a id="221" class="Symbol">{</a><a id="222" href="Categories.Diagram.Coend.html#222" class="Bound">D</a> <a id="224" class="Symbol">:</a> <a id="226" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="235" href="Categories.Diagram.Coend.html#190" class="Bound">o′</a> <a id="238" href="Categories.Diagram.Coend.html#193" class="Bound">ℓ′</a> <a id="241" href="Categories.Diagram.Coend.html#196" class="Bound">e′</a><a id="243" class="Symbol">}</a>
<a id="247" class="Symbol">(</a><a id="248" href="Categories.Diagram.Coend.html#248" class="Bound">F</a> <a id="250" class="Symbol">:</a> <a id="252" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="262" class="Symbol">(</a><a id="263" href="Categories.Category.Core.html#3132" class="Function">Category.op</a> <a id="275" href="Categories.Diagram.Coend.html#201" class="Bound">C</a><a id="276" class="Symbol">)</a> <a id="278" href="Categories.Diagram.Coend.html#201" class="Bound">C</a> <a id="280" href="Categories.Diagram.Coend.html#222" class="Bound">D</a><a id="281" class="Symbol">)</a> <a id="283" class="Keyword">where</a>
<a id="290" class="Keyword">private</a>
<a id="300" class="Keyword">module</a> <a id="C"></a><a id="307" href="Categories.Diagram.Coend.html#307" class="Module">C</a> <a id="309" class="Symbol">=</a> <a id="311" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="320" href="Categories.Diagram.Coend.html#201" class="Bound">C</a>
<a id="324" class="Keyword">module</a> <a id="D"></a><a id="331" href="Categories.Diagram.Coend.html#331" class="Module">D</a> <a id="333" class="Symbol">=</a> <a id="335" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="344" href="Categories.Diagram.Coend.html#222" class="Bound">D</a>
<a id="348" class="Keyword">open</a> <a id="353" href="Categories.Diagram.Coend.html#331" class="Module">D</a>
<a id="357" class="Keyword">open</a> <a id="362" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="377" class="Keyword">open</a> <a id="382" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="390" class="Keyword">variable</a>
<a id="403" href="Categories.Diagram.Coend.html#403" class="Generalizable">A</a> <a id="405" href="Categories.Diagram.Coend.html#405" class="Generalizable">B</a> <a id="407" class="Symbol">:</a> <a id="409" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="417" href="Categories.Diagram.Coend.html#417" class="Generalizable">f</a> <a id="419" href="Categories.Diagram.Coend.html#419" class="Generalizable">g</a> <a id="421" class="Symbol">:</a> <a id="423" href="Categories.Diagram.Coend.html#403" class="Generalizable">A</a> <a id="425" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="427" href="Categories.Diagram.Coend.html#405" class="Generalizable">B</a>
<a id="430" class="Keyword">open</a> <a id="435" class="Keyword">import</a> <a id="442" href="Level.html" class="Module">Level</a>
<a id="449" class="Keyword">open</a> <a id="454" class="Keyword">import</a> <a id="461" href="Categories.Diagram.Cowedge.html" class="Module">Categories.Diagram.Cowedge</a> <a id="488" href="Categories.Diagram.Coend.html#248" class="Bound">F</a>
<a id="490" class="Keyword">open</a> <a id="495" class="Keyword">import</a> <a id="502" href="Categories.Functor.html" class="Module">Categories.Functor</a>
<a id="521" class="Keyword">open</a> <a id="526" class="Keyword">import</a> <a id="533" href="Categories.Functor.Construction.Constant.html" class="Module">Categories.Functor.Construction.Constant</a>
<a id="574" class="Keyword">open</a> <a id="579" class="Keyword">import</a> <a id="586" href="Categories.NaturalTransformation.Dinatural.html" class="Module">Categories.NaturalTransformation.Dinatural</a>
<a id="629" class="Keyword">open</a> <a id="634" class="Keyword">import</a> <a id="641" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="671" href="Categories.Diagram.Coend.html#222" class="Bound">D</a>
<a id="674" class="Keyword">open</a> <a id="679" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="687" href="Categories.Diagram.Coend.html#248" class="Bound">F</a>
<a id="690" class="Keyword">record</a> <a id="Coend"></a><a id="697" href="Categories.Diagram.Coend.html#697" class="Record">Coend</a> <a id="703" class="Symbol">:</a> <a id="705" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="709" class="Symbol">(</a><a id="710" href="Level.html#602" class="Function">levelOfTerm</a> <a id="722" href="Categories.Diagram.Coend.html#248" class="Bound">F</a><a id="723" class="Symbol">)</a> <a id="725" class="Keyword">where</a>
<a id="733" class="Keyword">field</a>
<a id="Coend.cowedge"></a><a id="743" href="Categories.Diagram.Coend.html#743" class="Field">cowedge</a> <a id="751" class="Symbol">:</a> <a id="753" href="Categories.Diagram.Cowedge.html#596" class="Record">Cowedge</a>
<a id="764" class="Keyword">module</a> <a id="Coend.cowedge"></a><a id="771" href="Categories.Diagram.Coend.html#771" class="Module">cowedge</a> <a id="779" class="Symbol">=</a> <a id="781" href="Categories.Diagram.Cowedge.html#596" class="Module">Cowedge</a> <a id="789" href="Categories.Diagram.Coend.html#743" class="Field">cowedge</a>
<a id="799" class="Keyword">open</a> <a id="804" href="Categories.Diagram.Coend.html#771" class="Module">cowedge</a> <a id="812" class="Keyword">public</a>
<a id="821" class="Keyword">open</a> <a id="826" href="Categories.Diagram.Cowedge.html#596" class="Module">Cowedge</a>
<a id="837" class="Keyword">field</a>
<a id="Coend.factor"></a><a id="847" href="Categories.Diagram.Coend.html#847" class="Field">factor</a> <a id="857" class="Symbol">:</a> <a id="859" class="Symbol">(</a><a id="860" href="Categories.Diagram.Coend.html#860" class="Bound">W</a> <a id="862" class="Symbol">:</a> <a id="864" href="Categories.Diagram.Cowedge.html#596" class="Record">Cowedge</a><a id="871" class="Symbol">)</a> <a id="873" class="Symbol">→</a> <a id="875" href="Categories.Diagram.Cowedge.html#644" class="Function">cowedge.E</a> <a id="885" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="887" href="Categories.Diagram.Cowedge.html#644" class="Field">E</a> <a id="889" href="Categories.Diagram.Coend.html#860" class="Bound">W</a>
<a id="Coend.universal"></a><a id="895" href="Categories.Diagram.Coend.html#895" class="Field">universal</a> <a id="905" class="Symbol">:</a> <a id="907" class="Symbol">∀</a> <a id="909" class="Symbol">{</a><a id="910" href="Categories.Diagram.Coend.html#910" class="Bound">W</a> <a id="912" class="Symbol">:</a> <a id="914" href="Categories.Diagram.Cowedge.html#596" class="Record">Cowedge</a><a id="921" class="Symbol">}</a> <a id="923" class="Symbol">{</a><a id="924" href="Categories.Diagram.Coend.html#924" class="Bound">A</a><a id="925" class="Symbol">}</a> <a id="927" class="Symbol">→</a> <a id="929" href="Categories.Diagram.Coend.html#847" class="Field">factor</a> <a id="936" href="Categories.Diagram.Coend.html#910" class="Bound">W</a> <a id="938" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="940" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">cowedge.dinatural.α</a> <a id="960" href="Categories.Diagram.Coend.html#924" class="Bound">A</a> <a id="962" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="964" href="Categories.NaturalTransformation.Dinatural.html#851" class="Field">dinatural.α</a> <a id="976" href="Categories.Diagram.Coend.html#910" class="Bound">W</a> <a id="978" href="Categories.Diagram.Coend.html#924" class="Bound">A</a>
<a id="Coend.unique"></a><a id="984" href="Categories.Diagram.Coend.html#984" class="Field">unique</a> <a id="994" class="Symbol">:</a> <a id="996" class="Symbol">∀</a> <a id="998" class="Symbol">{</a><a id="999" href="Categories.Diagram.Coend.html#999" class="Bound">W</a> <a id="1001" class="Symbol">:</a> <a id="1003" href="Categories.Diagram.Cowedge.html#596" class="Record">Cowedge</a><a id="1010" class="Symbol">}</a> <a id="1012" class="Symbol">{</a><a id="1013" href="Categories.Diagram.Coend.html#1013" class="Bound">g</a> <a id="1015" class="Symbol">:</a> <a id="1017" href="Categories.Diagram.Cowedge.html#644" class="Function">cowedge.E</a> <a id="1027" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1029" href="Categories.Diagram.Cowedge.html#644" class="Field">E</a> <a id="1031" href="Categories.Diagram.Coend.html#999" class="Bound">W</a><a id="1032" class="Symbol">}</a> <a id="1034" class="Symbol">→</a> <a id="1036" class="Symbol">(∀</a> <a id="1039" class="Symbol">{</a><a id="1040" href="Categories.Diagram.Coend.html#1040" class="Bound">A</a><a id="1041" class="Symbol">}</a> <a id="1043" class="Symbol">→</a> <a id="1045" href="Categories.Diagram.Coend.html#1013" class="Bound">g</a> <a id="1047" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1049" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">cowedge.dinatural.α</a> <a id="1069" href="Categories.Diagram.Coend.html#1040" class="Bound">A</a> <a id="1071" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1073" href="Categories.NaturalTransformation.Dinatural.html#851" class="Field">dinatural.α</a> <a id="1085" href="Categories.Diagram.Coend.html#999" class="Bound">W</a> <a id="1087" href="Categories.Diagram.Coend.html#1040" class="Bound">A</a><a id="1088" class="Symbol">)</a> <a id="1090" class="Symbol">→</a> <a id="1092" href="Categories.Diagram.Coend.html#847" class="Field">factor</a> <a id="1099" href="Categories.Diagram.Coend.html#999" class="Bound">W</a> <a id="1101" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1103" href="Categories.Diagram.Coend.html#1013" class="Bound">g</a>
<a id="Coend.η-id"></a><a id="1108" href="Categories.Diagram.Coend.html#1108" class="Function">η-id</a> <a id="1113" class="Symbol">:</a> <a id="1115" href="Categories.Diagram.Coend.html#847" class="Field">factor</a> <a id="1122" href="Categories.Diagram.Coend.html#743" class="Field">cowedge</a> <a id="1130" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1132" href="Categories.Category.Core.html#630" class="Function">D.id</a>
<a id="1139" href="Categories.Diagram.Coend.html#1108" class="Function">η-id</a> <a id="1144" class="Symbol">=</a> <a id="1146" href="Categories.Diagram.Coend.html#984" class="Field">unique</a> <a id="1153" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a>
<a id="Coend.unique′"></a><a id="1166" href="Categories.Diagram.Coend.html#1166" class="Function">unique′</a> <a id="1174" class="Symbol">:(∀</a> <a id="1178" class="Symbol">{</a><a id="1179" href="Categories.Diagram.Coend.html#1179" class="Bound">A</a><a id="1180" class="Symbol">}</a> <a id="1182" class="Symbol">→</a> <a id="1184" href="Categories.Diagram.Coend.html#417" class="Generalizable">f</a> <a id="1186" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1188" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">cowedge.dinatural.α</a> <a id="1208" href="Categories.Diagram.Coend.html#1179" class="Bound">A</a> <a id="1210" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1212" href="Categories.Diagram.Coend.html#419" class="Generalizable">g</a> <a id="1214" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1216" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">cowedge.dinatural.α</a> <a id="1236" href="Categories.Diagram.Coend.html#1179" class="Bound">A</a><a id="1237" class="Symbol">)</a> <a id="1239" class="Symbol">→</a> <a id="1241" href="Categories.Diagram.Coend.html#417" class="Generalizable">f</a> <a id="1243" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1245" href="Categories.Diagram.Coend.html#419" class="Generalizable">g</a>
<a id="1249" href="Categories.Diagram.Coend.html#1166" class="Function">unique′</a> <a id="1257" class="Symbol">{</a><a id="1258" class="Argument">f</a> <a id="1260" class="Symbol">=</a> <a id="1262" href="Categories.Diagram.Coend.html#1262" class="Bound">f</a><a id="1263" class="Symbol">}</a> <a id="1265" class="Symbol">{</a><a id="1266" class="Argument">g</a> <a id="1268" class="Symbol">=</a> <a id="1270" href="Categories.Diagram.Coend.html#1270" class="Bound">g</a><a id="1271" class="Symbol">}</a> <a id="1273" href="Categories.Diagram.Coend.html#1273" class="Bound">eq</a> <a id="1276" class="Symbol">=</a> <a id="1278" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1280" class="Symbol">(</a><a id="1281" href="Categories.Diagram.Coend.html#984" class="Field">unique</a> <a id="1288" class="Symbol">{</a><a id="1289" class="Argument">W</a> <a id="1291" class="Symbol">=</a> <a id="1293" href="Categories.Diagram.Cowedge.html#769" class="Function">Cowedge-∘</a> <a id="1303" href="Categories.Diagram.Coend.html#743" class="Field">cowedge</a> <a id="1311" href="Categories.Diagram.Coend.html#1262" class="Bound">f</a><a id="1312" class="Symbol">}</a> <a id="1314" href="Relation.Binary.Structures.html#1577" class="Function">refl</a><a id="1318" class="Symbol">)</a> <a id="1320" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1322" href="Categories.Diagram.Coend.html#984" class="Field">unique</a> <a id="1329" class="Symbol">(</a><a id="1330" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1332" href="Categories.Diagram.Coend.html#1273" class="Bound">eq</a><a id="1334" class="Symbol">)</a>
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